The midfielder came off the bench for Gareth Bale on the hour and made his mark within 12 minutes.
Picking the ball up left of centre, Gylfi turned Phillipe Senderos on the edge of the box and then squared for Jermain Defoe to finish from eight yards.
That made it 2-0 after Sandro's opener and Jermain's second goal wrapped it up minutes later.
Gylfi curled another effort at goal and typical of the team's performance, was back in our penalty area to head away a free-kick in injury time.
"When I came on it was 1-0 and I wanted to help us finish the game off," said Gylfi.
"I managed to get the assist and was delighted to see Jermain score. That made it 2-0 and a little more comfortable. Jermain then scored the third.
"It was a very good win and great to keep a clean sheet away from home against a team like Fulham. That's three wins in a row and things are looking good."
SPURS v PANATHINAIKOS, EUROPA LEAGUE, WHL, THURSDAY DECEMBER 6
Tickets are currently on sale to all One Hotspur Members who are able to purchase a maximum of 2 tickets per membership.
Tickets now on RESTRICTED GENERAL SALE. In order for supporters to be able to purchase tickets they must have an account on our ticketing database as of 31 October 2012.
Ticket prices for the Panathinaikos fixture in all areas are priced as follows:
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